Method and apparatus for optically analyzing a surface

ABSTRACT

Apparatus and methods are provided for analyzing surface characteristics of a test object using broadband scanning interferometry. Test objects amenable to these apparatus and methods include but are not limited to semiconductor wafers, semiconductor devices, metallic surfaces, and the like. An interferometry system is used to obtain an interferometry signal and related to data embodied in the signal representative of the test object surface. This signal and/or data is used to construct an n-dimensional function that includes an independent frequency variable and an independent time variable, and/or an n-dimensional function that includes an independent scale variable and an independent time variable, and/or a multi-domain function. These functions are compared with various models to obtain a best match that is then used to characterize the test object surface.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to interferometric devices and methods foroptically analyzing surfaces, for example, such as the surfaces ofsemiconductor wafers, semiconductor devices, magnetic surfaces, and thelike.

2. Description of the Related Art

Interferometers are commonly used for testing the shape and topographyof surfaces. Good interferometers have height sensitivity in thesub-nanometer range, providing two-dimensional maps in rapid,non-contact operation. A large variety of optical interferometerconfigurations exist, as well as processing techniques to derive surfacetopography maps from the measured interferogram intensities.

The fringes of the interferogram are governed by the phase differencebetween a test beam and a reference beam. The phase of the test beamincludes a propagation phase delay and a phase shift on reflection atthe surface of a test object or test piece. While the propagation phaseis indicative of the test surface height, the phase shift on reflectionis affected by the material properties, composition, and surfacefeatures of the test piece. Although the variation of the phase shift onreflection is usually considered to be a source of error in the surfacetopography measurement, additional information about these test objectproperties carried in the interferograms can be extracted and exploited.

For test surfaces with uniform properties, the phase shift on reflectionis constant across the interferogram and only contributes to a constantheight offset, like the position alignment of the test piece. It doesnot affect the surface topography measurement. Examples include adielectric glass surface, where the phase shift on reflection is pi (π)radians, or an aluminized mirror surface, where the phase shift isdifferent from π but uniform in the interferogram

For non-uniform test surfaces, the phase shift on reflection variesacross the interferogram and hence affects the topography measurements.Examples include read-write head surfaces in magnetic hard disks, wherepart of the test surface is a dielectric and another part is metallic,or a patterned semiconductor wafer surface in the copperchemical-mechanical-polishing (“CMP”) process, where part of the testsurface consists of metallic copper and other parts consist ofdielectric multilayer film stacks on a silicon wafer.

In integrated circuit manufacturing, the semiconductor switches in thewafer material need to be connected by conductive wiring to obtainfunctioning circuits. This wiring structure is manufactured in layers bydepositing a metal overcoat on the insulating layers that are patternedwith vias and trenches, and then removing all excess metal material byCMP. Each metal wiring layer can contain several dielectric insulatinglayers of different materials such that, in general, the top surfaceincludes metal areas and areas with complex multilayer film stacks.

After the CMP process, only the vias and trenches are filled with metal,thus functioning as conductive paths. Currently, tungsten is commonlyused for the first metal layer over the silicon, while higher metallayers mainly use copper. The CMP process exhibits material- andgeometry-dependent removal rates, leading to difficulties in obtainingan ideally flat top surface for metal and insulator. One maincharacteristic of the top surface topography is called “dishing,” whichrefers to the surface height difference by which the metal surface islower than the surrounding insulator. Another characteristic is called“erosion,” which refers to the surface height difference by whichinsulator material interspersed with metal is lower than solidinsulator. Another item of interest in the manufacturing process is thefilm thickness of the dielectric layers as well as of the metal lines.Furthermore, the width, shape, sidewall angle, or in general themicro-structure, of the fine metal lines frequently are of interest.

Currently, the top surface topography of CMP products is measured withtactile stylus tools or atomic-force profilers. The contacting nature ofthese profilers restricts their usage to special targets in the scribeline between chips. Furthermore, the measurement process is slow becauseit is inherently a single-point measurement where profiles ortwo-dimensional maps have to be built up sequentially. The filmthickness is measured using spectrophotometers or ellipsometers, whichare unable to provide information about the surface topography. Thewidth and shape of fine metal lines, or in general of micro-structures,are measured by atomic-force profilers or scatterometers.

U.S. Pat. No. 6,545,763 to Kim et al. discloses a method for themeasurement of surfaces with thin films based on white-light scanninginterferometry which extracts a spectral phase distribution from thetemporal intensity or time-domain signal by Fourier transform. Thespectral phase distribution is then compared with a theoreticallygenerated phase distribution by modeling the measured surface withdifferent properties, such as film thicknesses, refractive indexes, etc.Once the best match is found, the surface properties as well as thesurface topography are determined. Difficulties can arise with thinnerfilms using this approach.

The available technology has been limited and in some senses deficient,for example, in that known apparatus and methods are unable to providefast, non-contact surface topography measurements in the presence ofvarying phase shifts on reflection. The need also remains for apparatusand methods that can quickly and efficiently measure surface topographyas well as surface composition. The term “surface composition” as usedherein, broadly construed, means not only the chemical or materialcomposition at the surface or top plane of the object, but otherproperties and characteristics at or below the surface. It can include,for example, film layers lying at or beneath the surface or top plane ofthe material, optical properties such as refractive index, surfacemicrostructure, electrical properties such as conductivity orresistivity, and the like.

OBJECTS OF THE INVENTION

Accordingly, an object of the present invention is to provide apparatusand methods that enable non-contact analysis of surfaces quickly andefficiently.

Another object of the present invention is to provide apparatus andmethods that effectively and accurately analyze the characteristics of atest object surface.

Another object of the present invention is to provide apparatus andmethods that enable one to efficiently analyze characteristics of a testobject surface, for example, by reducing the processing demands whileobtaining accurate and reliable results.

Additional objects and advantages of the invention will be set forth inthe description which follows, and in part will be apparent from thedescription, or may be learned by practice of the invention. The objectsand advantages of the invention may be realized and obtained by means ofthe instrumentalities and combinations pointed out in the appendedclaims.

SUMMARY OF THE INVENTION

To achieve the foregoing objects, and in accordance with the purposes ofthe invention as embodied and broadly described in this document, amethod is provided according to one aspect of the invention foranalyzing surface characteristics of a test object using broadband lightscanning interferometry. The method comprises acquiring data from ascanning interferometry signal from a surface location of a test object,transforming the acquired data to an n-dimensional function comprisingan independent frequency variable and an independent time variable, andcomparing the transformed n-dimensional function with a set of models todetermine a best match from the comparison.

In presently preferred embodiments, n equals 2. In such embodiments, thescanning interferometer may comprise an illuminating numerical apertureand an imaging numerical aperture, and the acquiring of the data maycomprise using an illuminating numerical aperture that is smaller thanthe imaging numerical aperture. Preferably, the illuminating numericalaperture is less than about 0.15.

It is preferred that the transforming of the acquired data to then-dimensional function comprises creating a spectrogram modified by aweighting function corresponding to desired parameters of interest. Italso is preferred that the transforming of the acquired data to then-dimensional function comprises using a time-frequency transform. Thetransforming of the acquired data to the n-dimensional function maycomprise using a Gabor transform.

It also is preferred that the transforming of the acquired data to then-dimensional function comprises applying a window function to theinterferometry signal. The application of the window function preferablycomprises applying multiple windows to the interferometry signal, andmore preferably sequentially over time.

The method also preferably comprises dividing the time domain signalinto a plurality of time-differentiated segments, and dividing thefrequency domain signal into a plurality of time differentiated segmentscorresponding to the plurality of time differentiated segments of thetime domain signal, and the transforming of the acquired data into then-dimensional function preferably comprises using the plurality of timedifferentiated segments of the frequency domain signal and the pluralityof time differentiated segments of the time domain signal to constructthe n-dimensional function as a function of time. The plurality of timedifferentiated segments of the time domain signal may overlap oneanother. The plurality of time differentiated segments of the frequencydomain signal also may overlap one another.

In preferred implementations of this method, the acquiring of the datafrom the scanning interferometry signal may comprise spectral shaping ofthe interferometric signal. The acquiring of the data from the scanninginterferometry signal thus may comprise using light having an originallight spectrum, and the method may further comprise modifying theoriginal light spectrum to facilitate the comparing of the n-dimensionalfunction with the set of models to determine the best match from thecomparison. The modification of the original light spectrum also maycomprise using optical components to modify the original light spectrum.The modification of the original light spectrum also may comprise usingan optical filter comprising a selected transmittance function. Themodification of the original light spectrum also may comprise using aprogrammable filter.

The method according to this aspect of the invention preferablycomprises using a processor to modify the interferometry signal tofacilitate the comparing of the n-dimensional function with the models.

In accordance with another aspect of the invention, a method is providedfor analyzing surface characteristics of a test object using broadbandlight scanning interferometry. The method comprises acquiring data froma scanning interferometry signal from a surface location of a testobject, transforming the acquired data to an n-dimensional functionhaving an independent scale variable and an independent time variable,and comparing the transformed n-dimensional function with a set ofstored n-dimensional function models to determine a best match from thecomparison.

In presently preferred implementations of this method, n equals 2. Alsoin such implementations, the scanning interferometer may comprise anilluminating numerical aperture and an imaging numerical aperture, andthe acquiring of the data may comprise using an illuminating numericalaperture that is smaller than the imaging numerical aperture. Theilluminating numerical aperture preferably is less than about 0.15.

In presently preferred implementations, the transforming of the acquireddata to the n-dimensional function preferably comprises creating ascalogram modified by a weighting function corresponding to desiredparameters of interest. The transforming of the. acquired data to then-dimensional function also may comprise using a time-scale transform.The transforming of the. acquired data to the n-dimensional functionpreferably comprises using a wavelet transform.

The transforming of the acquired data to the n-dimensional function alsopreferably comprises applying a window function to the interferometrysignal. The application of the window function may and preferably doescomprise applying multiple windows to the interferometry signal. Theapplication of the multiple windows also preferably comprises applyingthe multiple windows sequentially over time.

In presently preferred implementations, the method further comprisesdividing the time domain signal into a plurality of time-differentiatedsegments, and dividing the scale domain signal into a plurality of timedifferentiated segments corresponding to the plurality of timedifferentiated segments of the time domain signal, and the transformingof the acquired data into the n-dimensional function comprises using theplurality of time differentiated segments of the scale domain signal andthe plurality of time differentiated segments of the time domain signalto construct the n-dimensional function as a function of time. Theplurality of time differentiated segments of the time domain signal mayoverlap one another, as may the plurality of time differentiatedsegments of the scale domain signal.

In presently preferred implementations of the method according to thisaspect of the invention, the acquiring of the data from the scanninginterferometry signal may comprise spectral shaping of theinterferometry signal. Thus, the acquiring of the data from the scanninginterferometry signal may comprise using light having an original lightspectrum, and the method further may comprise modifying the originallight spectrum to facilitate the comparing of the n-dimensional functionwith the set of stored models. The modification of the original lightspectrum may comprise using an optical filter comprising a selectedtransmittance function. The optical filter may comprise a programmablefilter. Preferred implementations of this method may comprise using aprocessor to modify the interferometry signal to facilitate thecomparing of the n-dimensional function with the models.

In accordance with another aspect of the invention, a method is providedfor analyzing a surface of a test object. The method comprises using aninterferometry system comprising a broadband light beam to scan thesurface of the test object and to thereby generate an interferometrysignal comprising a time domain signal, applying a window function tothe interferometry signal to obtain a windowed interferometry signal,obtaining a frequency domain signal from the windowed interferometrysignal, and constructing a multi-domain function from the frequencydomain signal and the time domain signal.

In presently preferred implementations of the method according to thisaspect of the invention, the multi-domain function comprises a 2-domainfunction. In such implementations, the application of the windowfunction preferably comprises applying multiple windows to theinterferometry signal. The application of the multiple windows comprisesapplying the multiple windows sequentially over time.

The method preferably comprises using of the interferometry system toscan the surface of the test object, which preferably comprisesvertically scanning a position on the surface of the test object andgenerating the interferometry signal and the time domain signal atselected unique times during the vertical scanning, and the dividing ofthe time domain signal into the plurality of time-differentiatedsegments. The time-differentiated segments of the plurality oftime-differentiated segments preferably are equally spaced from oneanother in time. The dividing of the time domain signal into theplurality of time-differentiated segments preferably comprises dividingthe vertical scan into a corresponding plurality of vertical scansegments, wherein the vertical scan segments are of equal length.

The method according to this aspect of the invention preferably furthercomprises dividing the time domain signal into a plurality oftime-differentiated segments, and dividing the frequency domain signalinto a plurality of time differentiated segments corresponding to theplurality of time differentiated segments of the time domain signal,wherein the construction of the multi-domain function comprises usingthe plurality of time differentiated segments of the frequency domainsignal and the plurality of time differentiated segments of the timedomain signal to construct the multi-domain function as a function oftime. The plurality of time differentiated segments of the time domainsignal preferably overlap one another, and the plurality of timedifferentiated segments of the frequency domain signal also overlap oneanother.

Methods according to this aspect of the invention also preferablycomprise comparing the multi-domain function with a set of models todetermine a best match from the comparison.

The test object in many instances may be assumed to comprise m toplayers and n bottom layer. In such instances, such methods alsopreferably comprise the use of the interferometry system to cause thebroadband light beam to comprise a spectral composition that facilitatesobtaining the frequency domain signal for the m top layers of the testobject while disfavoring obtaining the frequency domain signal for the nbottom layers of the test object. The use of the interferometry systemalso may comprise causing the broadband light beam to comprise aspectral composition such that the frequency domain signal for the m toplayers of the test object is distinguishable from the frequency domainsignal for the n bottom layers of the test object.

The broadband light beam typically will comprise amplitude components,in which case the causing of the broadband light beam to comprise aspectral composition that facilitates obtaining the frequency domainsignal for the m top layers of the test object while disfavoringobtaining the frequency domain signal for the n bottom layers of thetest object may comprise modifying the spectral composition based on theamplitude components. The use of the interferometry system to cause thebroadband light beam to comprise a spectral composition that facilitatesobtaining the frequency domain signal for the m top layers of the testobject while disfavoring obtaining the frequency domain signal for the nbottom layers of the test object also may comprise causing the broadbandlight source to have an original light spectrum that includes thespectral composition. The use of the interferometry system to cause thebroadband light beam to comprise a spectral composition that facilitatesobtaining the frequency domain signal for the m top layers of the testobject while disfavoring obtaining the frequency domain signal for the nbottom layers of the test object also may comprise using spectralshaping optics positioned within the broadband light beam to spectrallyshape the broadband light beam to obtain the spectral composition. Theuse of the interferometry system to cause the broadband light beam tocomprise the spectral composition that facilitates obtaining thefrequency domain signal for the m top layers of the test object whiledisfavoring obtaining the frequency domain signal for the n bottomlayers of the test object also may comprise using means positionedwithin the broadband light beam for spectrally shaping the broadbandlight beam to obtain the spectral components.

Where the test object comprises m top layers and n bottom layers, theobtaining of the frequency domain signal from the interferometry signalmay comprise using a processor to spectrally shape the interferometrysignal to comprise a spectral composition that facilitates obtaining thefrequency domain signal for the m top layers of the test object whiledisfavoring obtaining the frequency domain signal for the n bottomlayers of the test object. The obtaining of the frequency domain signalfrom the interferometry signal also may comprise using a processor tocause the interferometry signal to comprise a spectral composition suchthat the frequency domain signal for the m top layers of the test objectis distinguishable from the frequency domain signal for the n bottomlayers of the test object.

In accordance with another aspect of the invention, an apparatus isprovided for analyzing surface characteristics of a test object. Theapparatus comprises a scanning interferometry system for generating aninterferometry signal from a surface location of a test object, and aprocessor that transforms the acquired data to an n-dimensional functioncomprising an independent frequency variable and an independent timevariable, and that compares the transformed n-dimensional function witha set of models to determine a best match from the comparison. Inaccordance with another aspect of the invention, an apparatus isprovided for analyzing surface characteristics of a test object. Theapparatus comprises a scanning interferometry system that generates ascanning interferometry signal from a surface location of a test object,and a processor that transforms the acquired data to an n-dimensionalfunction having an independent scale variable and an independent timevariable, and that compares the transformed n-dimensional function witha set of stored n-dimensional function models to determine a best matchfrom the comparison.

In accordance with another aspect of the invention, an apparatus isprovided for analyzing a surface of a test object. The apparatuscomprises a scanning interferometry system that generates aninterferometry signal from the surface of the test object, wherein theinterferometry signal comprises a time domain signal, and a processorthat receives the interferometry signal, uses the interferometry signalto generate a frequency domain signal, and uses the time domain signaland the frequency domain signal to construct a multi-domain functionthat comprises information from the frequency domain signal and the timedomain signal.

In presently preferred embodiments of each of these aspects of theinvention, n preferably equals 2.

In such embodiments, the processor preferably further comprises meansfor applying a window function to the interferometry signal, and morepreferably the processor further comprises means for applying a windowfunction comprising multiple windows to the interferometry signal. Theprocessor preferably further comprises means for applying a windowfunction comprising multiple windows, wherein the multiple windows areapplied sequentially over time.

In presently preferred embodiments, the scanning interferometry systemcomprises means for vertically scanning a position on the surface of thetest object, and the processor comprises means for dividing the timedomain signal into a plurality of time-differentiated segments anddividing the frequency domain signal into a plurality of timedifferentiated segments corresponding to the plurality of timedifferentiated segments of the time domain signal, and functiongenerating means for using the plurality of time differentiated segmentsof the frequency domain signal and the plurality of time differentiatedsegments of the time domain signal to construct the multi-domainfunction as a function of time. Preferably, the means for dividing thetime domain signal into a plurality of time-differentiated segments anddividing the frequency domain signal into a plurality of timedifferentiated segments spaces the time-differentiated segments so thateach of the time differentiated segments has an equal time duration. Themeans for dividing the time domain signal into a plurality oftime-differentiated segments and dividing the frequency domain signalinto a plurality of time differentiated segments preferably spaces thetime-differentiated segments so that each of the time differentiatedsegments has an equal length. The means for dividing the time domainsignal into a plurality of time-differentiated segments and dividing thefrequency domain signal into a plurality of time differentiated segmentsalso preferably divides the plurality of time differentiated segments ofthe time domain signal so that the time differentiated segments overlapone another. In addition, it is preferred that the means for dividingthe time domain signal into a plurality of time-differentiated segmentsand dividing the frequency domain signal into a plurality of timedifferentiated segments divides the plurality of time differentiatedsegments of the frequency domain signal so that the time differentiatedsegments of the frequency domain signal overlap one another.

In presently preferred embodiments of these apparatus, and where thetest object can be assumed to comprise m top layers and n bottom layers,the processor comprises means for spectrally shaping the multi-spectrallight beam to cause the interferometry signal to comprise a spectralcomposition that facilitates obtaining the frequency domain signal forthe m top layers of the test object while disfavoring obtaining thefrequency domain signal for the n bottom layers of the test object.Similarly, in each such embodiment, the processor may comprise means forcausing the interferometry signal to comprise a spectral compositionsuch that the frequency domain signal for the m top layers of the testobject is distinguishable from the frequency domain signal for the nbottom layers of the test object.

In presently preferred embodiments of these apparatus, the processorpreferably comprises means for comparing the n-dimensional function witha set of n-dimensional function models to determine a best match fromthe comparison. The processor preferably comprises comparing themulti-domain function with a set of multi-domain function models todetermine a best match from the comparison.

Such apparatus also may comprise a display operatively coupled to theprocessor that displays the output as a three dimensional graphcomprising a first orthogonal axis representing the time domain signal,a second orthogonal axis representing the frequency domain signal, and athird orthogonal axis representing an amplitude of the n-dimensionaland/or multi-domain signal.

Such apparatus also may comprise a storage device for storing aplurality of models corresponding respectively to a plurality of testobject conditions.

In accordance with another aspect of the invention, an apparatus isprovided for analyzing a surface of a test object. The apparatuscomprises means for acquiring data from a scanning interferometry signalfrom a surface location of a test object, and means for transforming theacquired data to an n-dimensional function comprising an independentfrequency variable and an independent time variable, and means forcomparing the transformed n-dimensional finction with a set of models todetermine a best match from the comparison.

In accordance with another aspect of the invention, an apparatus isprovided for analyzing surface characteristics of a test object. Theapparatus comprises means for acquiring data from a scanninginterferometry signal from a surface location of a test object, meansfor transforming the acquired data to an n-dimensional function havingan independent scale variable and an independent time variable, andmeans for comparing the transformed n-dimensional function with a set ofn-dimensional function models to determine a best match from thecomparison.

In accordance with still another aspect of the invention, an apparatusis provided for analyzing a surface of a test object, wherein theapparatus comprises means for generating an interferometry signal fromthe surface of the test object, wherein the interferometry signalcomprises a time domain signal, and means for using the interferometrysignal to generate a frequency domain signal and for generating amulti-domain function using the time domain signal and the frequencydomain signal.

In each of the aforementioned apparatus, it is preferred that n equals2.

The means for using the interferometry signal to generate a frequencydomain signal and for generating a multi-domain function using the timedomain signal and the frequency domain signal further may comprise meansfor applying a window function to the interferometry signal. The meansfor using the interferometry signal to generate a frequency domainsignal and for generating a multi-domain function using the time domainsignal and the frequency domain signal also further may comprise meansfor applying a window function comprising multiple windows to theinterferometry signal. The means for using the interferometry signal togenerate a frequency domain signal and for generating a multi-domainfunction using the time domain signal and the frequency domain signalalso may further comprise means for applying a window functioncomprising multiple windows, wherein the multiple windows are appliedsequentially over time.

The means for using the interferometry signal to generate a frequencydomain signal and for generating a multi-domain function using the timedomain signal and the frequency domain signal preferably furthercomprises means for dividing the time domain signal into a plurality oftime-differentiated segments and for dividing the frequency domainsignal into a plurality of time differentiated segments corresponding tothe plurality of time differentiated segments of the time domain signal,and means for generating the multi-domain function using the pluralityof time differentiated segments of the frequency domain signal and theplurality of time differentiated segments of the time domain signal sothat the multi-domain is a function of time.

Each of the aforementioned means for processing preferably comprise acomputer such as a commercially available general purpose computer,programmed to carry out the functions described herein above. Othermeans of course may be used, for example, such as a dedicated processor,a system or network of computers, and the like.

Each of these apparatus also may and preferably do comprise means forcomparing the multi-domain function to a plurality of models to select abest match of the models with the multi-domain function. Such means alsopreferably comprise a general purpose computer with appropriateprogramming as described herein above, or the like.

Each of these apparatus further may comprises display means fordisplaying the multi-domain function as a three dimensional graphcomprising a first orthogonal axis representing the time domain signal,a second orthogonal axis representing the frequency domain signal, and athird orthogonal axis representing an amplitude of the multi-domainfunction. They also preferably comprise means for storing a plurality ofmodels corresponding respectively to a plurality of test objectconditions.

In accordance with another aspect of the invention, an apparatus isprovided for processing an interferometry signal from an analysis of atest object by a scanning interferometry system. The apparatus accordingto this aspect of the invention comprises a processor that transformsthe acquired data to an n-dimensional function comprising an independentfrequency variable and an independent time variable, and that comparesthe transformed n-dimensional function with a set of models to determinea best match from the comparison.

In accordance with a related aspect of the invention, an apparatus isprovided for processing an interferometry signal from an analysis of atest object by a scanning interferometry system. The apparatus comprisesa processor that transforms the acquired data to an n-dimensionalfunction having an independent scale variable and an independent timevariable, and that compares the transformed n-dimensional function witha set of stored n-dimensional function models to determine a best matchfrom the comparison.

In accordance with another related aspect of the invention, an apparatusis provided for processing an interferometry signal from an analysis ofa test object by a scanning interferometry system, wherein theinterferometry signal comprises a time domain signal. The apparatuscomprises a processor that uses the interferometry signal to generate afrequency domain signal and that uses the time domain signal and thefrequency domain signal to construct a multi-domain function thatcomprises information from the frequency domain signal and informationfrom the time domain signal.

The apparatus according to these aspects of the invention preferablyfurther comprise means for applying a window function to theinterferometry signal, and more preferably comprise means for applying awindow function comprising multiple windows to the interferometrysignal. Preferably they comprise applying the multiple windowssequentially over time.

Preferred embodiments of these apparatus also preferably comprise meansfor dividing the time domain signal into a plurality oftime-differentiated segments and dividing the frequency domain signalinto a plurality of time differentiated segments corresponding to theplurality of time differentiated segments of the time domain signal,means for using the plurality of time differentiated segments of thefrequency domain signal and the plurality of time differentiatedsegments of the time domain signal to construct the multi-dimensionalfunction as a function of time. The dividing means preferably spaces thetime-differentiated segments so that each of the time differentiatedsegments has an equal time duration. The dividing means also preferablyspaces the time-differentiated segments so that each of the timedifferentiated segments has an equal length. The dividing meanspreferably divides the plurality of time differentiated segments of thetime domain signal so that the time differentiated segments overlap oneanother, and so that the time differentiated segments of the frequencydomain signal overlap one another.

In each of these apparatus, the processor preferably comprises means forspectrally shaping the broadband light beam to cause the interferometrysignal to comprise a spectral composition that facilitates obtaining thefrequency domain signal for the m top layers of the test object whiledisfavoring obtaining the frequency domain signal for the n bottomlayers of the test object. The processor also preferably comprises meansfor causing the interferometry signal to comprise a spectral compositionsuch that the frequency domain signal for the m top layers of the testobject is distinguishable from the frequency domain signal for the nbottom layers of the test object.

These apparatus also preferably further comprise means for comparing themulti-domain function with a set of models to determine a best matchfrom the comparison. The output preferably comprises means foroutputting the multi-domain function as a three dimensional graphcomprising a first orthogonal axis representing the time domain signal,a second orthogonal axis representing the frequency domain signal, and athird orthogonal axis representing an amplitude of the multi-domainsignal.

Each of these aspects of the invention and the related means forprocessing and outputting the date, the means as described herein abovepreferably comprise a general purpose computer programmed to carry outthese functions, or similar processing devices as referred to hereinabove.

These apparatus preferably further comprise a display generator thatgenerates a display of the multi-domain function as a three dimensionalgraph that comprises a first orthogonal axis representing a timecomponent from the time domain signal, a second orthogonal axisrepresenting a frequency component from the frequency domain signal, anda third orthogonal axis representing an amplitude of the multi-domainsignal.

They also preferably comprise a storage device operatively coupled tothe processor for storing a plurality of models correspondingrespectively to a plurality of test object conditions, and the processorpreferably comprises comparing means, such as the general purposecomputer, for comparing the multi-domain function to the plurality ofmodels to select a best match. Preferably, the storage device stores atleast one parameter for each of the plurality of models, the processorcomprises means for generating at least one test object parameter forthe test object from the multi-domain function, and the comparing meanscompares the at least one parameter from the multi-dimensional functionto at least one parameter of the plurality of models to select anoptimal one of the models.

In accordance with yet another aspect of the invention, a machinereadable medium is provided. The medium comprises a program that causesthe machine to use a scanning interferometry system to generate aninterferometry signal from a surface location of a test object, whereinthe program causes the machine to transform the acquired data to ann-dimensional function comprising an independent frequency variable andan independent time variable, and wherein the program causes the machineto compare the transformed n-dimensional function with a set of modelsto determine a best match from the comparison.

In accordance with a related aspect of the invention, a machine readablemedium is provided, wherein the medium comprises a program that causesthe machine to use a scanning interferometry system to generate ascanning interferometry signal from a surface location of a test object,wherein the program causes the machine to transform the acquired data toan n-dimensional function having an independent scale variable and anindependent time variable, and wherein the program causes the machine tocompare the transformed n-dimensional function with a set of storedn-dimensional function models to determine a best match from thecomparison.

In accordance with still another aspect of the invention, a mediumreadable by a machine is provided. The medium comprises a program thatcauses the machine to use an interferometry signal comprising a timedomain signal from a broadband interferometry system, whichinterferometry signal is obtained from a test surface, wherein theprogram causes the machine to use the interferometry signal to generatea frequency domain signal, wherein the program causes the machine to usethe time domain signal and the frequency domain signal to construct amulti-domain function that comprises information from the time domainsignal and the frequency domain signal, and wherein the program causesthe machine to use the multi-domain function to obtain informationuseful in characterizing the surface of the test object.

In presently preferred embodiments of each of these machine readablemedium aspects of the invention, n preferably is equal to 2.

In presently preferred embodiments of each of these aspects of theinvention, the program preferably further comprises means for causingthe machine to apply a window function to the interferometry signal. Theprogram also preferably further comprises means for causing the machineto apply a window function comprising multiple windows to theinterferometry signal. The program further may comprise means forcausing the machine to apply a window function comprising multiplewindows to the interferometry signal, wherein the multiple windows areapplied sequentially over time. Such means in both instances maycomprise computer code constructed to cause these functions to beperformed by a processor, such as the processor of a general purposecomputer. The program also may comprise means for causing the machine todivide the time domain signal into a plurality of time-differentiatedsegments and to divide the frequency domain signal into a plurality oftime differentiated segments corresponding to the plurality of timedifferentiated segments of the time domain signal, means for causing themachine to use the plurality of time differentiated segments of thefrequency domain signal and the plurality of time differentiatedsegments of the time domain signal to construct the multi-domainfunction as a function of time. Such means again may comprise computercode for causing the processor to perform these functions. The dividingmeans preferably spaces the time-differentiated segments so that each ofthe time differentiated segments has an equal time duration. Thedividing means also may cause the machine to space thetime-differentiated segments so that each of the time differentiatedsegments has an equal length. The dividing means also may cause themachine to divide the plurality of time differentiated segments of thetime domain signal so that the time differentiated segments overlap oneanother. The dividing means preferably causes the machine to divide theplurality of time differentiated segments of the frequency domain signalso that the time differentiated segments of the frequency domain signaloverlap one another.

The program also preferably comprises means for causing the machine tospectrally shape the broadband light beam to cause the interferometrysignal to comprise a spectral composition that facilitates obtaining thefrequency domain signal for the m top layers of the test object whiledisfavoring obtaining the frequency domain signal for the n bottomlayers of the test object. The program also may comprise means forcausing the machine to cause the interferometry signal to comprise aspectral composition such that the frequency domain signal for the m toplayers of the test object is distinguishable from the frequency domainsignal for the n bottom layers of the test object. These means maycomprise computer code that controls a device for filtering or otherwisemodifying the optics of the interferometry system, and/or computer codethat operates within the processor to modify the interferometry signalto accomplish these functions, and the like.

The program also preferably further comprises means for causing themachine to compare the multi-domain function with a set of models todetermine a best match from the comparison. These means also preferablycomprise computer code that causes the processor to perform thesefunctions.

The program also may comprise means for causing the machine to outputthe multi-domain function as a multi-domain graph and/or multi-domaindisplay. Such means also preferably comprise suitable computer code foroperating in conjunction with the processor to cause such output. Theprogram preferably comprises a first program component that generatesthe multi-domain display as a three dimensional graph that comprises afirst orthogonal axis representing a time component from the time domainsignal, a second orthogonal axis representing a frequency component fromthe frequency domain signal, and a third orthogonal axis representing anamplitude of the n-dimensional function and/or the multi-domainfunction. The program also preferably comprises a second programcomponent that compares the n-dimensional function and/or themulti-domain function to a plurality of models correspondingrespectively to a plurality of test object conditions and selects thebest one of the models.

In accordance with another aspect of the invention, a method is providedfor analyzing a surface of a test object. The method comprises using aninterferometry system comprising a broadband light beam to scan thesurface of the test object and to thereby generate an interferometrysignal comprising a time domain signal, applying a window function tothe interferometry signal to obtain a windowed interferometry signal,and characterizing the surface of the test object using the windowedinterferometry signal.

The application of the window function preferably comprises applyingmultiple windows to the interferometry signal. The application of themultiple windows comprises applying the multiple windows sequentiallyover time.

In presently preferred implementations of the method, the using of theinterferometry system to scan the surface of the test object comprisesvertically scanning a position on the surface of the test object andgenerating the interferometry signal and the time domain signal atselected unique times during the vertical scanning, and the dividing ofthe time domain signal into the plurality of time-differentiatedsegments. The time-differentiated segments of the plurality oftime-differentiated segments preferably are equally spaced from oneanother in time. The dividing of the time domain signal into theplurality of time-differentiated segments also preferably comprisesdividing the vertical scan into a corresponding plurality of verticalscan segments, wherein the vertical scan segments are of equal length.

The method also may further comprise dividing the time domain signalinto a plurality of time-differentiated segments, and dividing thefrequency domain signal into a plurality of time differentiated segmentscorresponding to the plurality of time differentiated segments of thetime domain signal, wherein the characterization of the surface of thetest object comprises using the plurality of time differentiatedsegments of the frequency domain signal and the plurality of timedifferentiated segments of the time domain signal to characterize thesurface of the test object.

The plurality of time differentiated segments of the time domain signalpreferably overlap one another, and the plurality of time differentiatedsegments of the frequency domain signal also preferably overlap oneanother.

In accordance with another aspect of the invention, an apparatus isprovided for analyzing a surface of a test object. The apparatuscomprises a scanning interferometry system that generates a windowedinterferometry signal from the surface of the test object.

The scanning interferometry system preferably generates the windowedinterferometry signal to comprise multiple windows. The scanninginterferometry system also preferably generates windows of the windowedinterferometry signal sequentially over time.

In accordance with another aspect of the invention, an apparatus isprovided for analyzing a surface of a test object. The apparatusaccording to this aspect of the invention comprises a scanninginterferometry system that generates an interferometry signal from thesurface of the test object, and a processor that applies a windowfunction to the interferometry signal.

The processor preferably applies the window function to comprisemultiple windows to the interferometry signal, and the processorpreferably applies each of the windows sequentially over time.

In presently preferred embodiments according to this aspect of theinvention, the scanning interferometry system comprises means forvertically scanning a position on the surface of the test object, andthe processor comprises means for dividing the interferometry signalinto a plurality of time-differentiated segments. The means forvertically scanning may comprise one or more devices for moving the testobject surface relative to the reference surface, directly or indirectlythe means for dividing the interferometry signal into the plurality oftime-differentiated segments may comprise framing hardware or softwareassociated with a camera, an associated processor, or both. The meansfor dividing the interferometry signal into a plurality oftime-differentiated segments preferably spaces the time-differentiatedsegments so that each of the time differentiated segments has an equaltime duration. The means for dividing the interferometry signal into aplurality of time-differentiated segments also preferably spaces thetime-differentiated segments so that each of the time differentiatedsegments has an equal length. The means for dividing the interferometrysignal into a plurality of time-differentiated segments also preferablydivides the plurality of time differentiated segments so that the timedifferentiated segments overlap one another.

These aspects of the invention can provide apparatus and methods thatenable one to determine the surface topography of structured surfaces inthe presence of varying phase shift on reflection. The phase shift onreflection may be due to material properties of bulk surfaces, single ormulti-layer film stacks on a substrate, and/or micro-structures on asubstrate or as part of a film stack. The invention also can provideapparatus and methods capable of simultaneously or separatelydetermining additional parameters of the test object, e.g., layerthickness and/or material refractive index for film stacks, or linewidth and structure depth of micro-structures.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of the specification, illustrate a presently preferred embodimentsand methods of the invention and, together with the general descriptiongiven above and the detailed description of the preferred embodimentsand methods given below, serve to explain the principles of theinvention. Of the drawings:

FIG. 1 is a block diagram of an interferometer according to a presentlypreferred embodiment of one aspect of the invention, and is used hereinto illustrate a presently preferred implementation of a method accordingto another aspect of the invention;

FIG. 2 is a detailed side view of an objective lens of a portion of theembodiment of FIG. 1, and shows light rays impinging upon and reflectingfrom a test piece;

FIG. 3 is a diagram of a spectral filter useable with the interferometerof FIG. 1 in accordance with another aspect of the invention;

FIG. 4 shows another spectral filter useable with the interferometer ofFIG. 1;

FIG. 5 shows still another spectral filter useable with theinterferometer of FIG. 4;

FIG. 6 is a graph of an interferometry signal outputted by theinterferometer of FIG. 1, wherein the intensity of the interferometricsignal is plotted along the y axis as a function of time on the x axis;

FIG. 7 is a graphical output according to a presently preferred methodimplementation according to another aspect of the invention, whereinboth time-domain and frequency domain information are presented for theinterferometry signal of FIG. 6, and in which time or distancecorresponds to the x axis, frequency corresponds to the y axis, and therelative amplitudes of the time-frequency domain signal are plotted asgrey level;

FIG. 8 is a graphical output according to another aspect of theinvention, wherein both time-domain and scale domain information arepresented for the interferometry signal of FIG. 6, and in which time ordistance corresponds to the x axis, scale corresponds to the y axis, andthe relative amplitudes of the time-scale domain signal are plotted asgrey level; and

FIG. 9 is a flow diagram that summarizes processing of an interferometrysignal from the interferometer of FIG. 1 according to yet another aspectof the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS AND METHODS

Reference will now be made in detail to the presently preferredembodiments and methods of the invention as illustrated in theaccompanying drawings, in which like reference characters designate likeor corresponding parts throughout the drawings. It should be noted,however, that the invention in its broader aspects is not limited to thespecific details, representative devices and methods, and illustrativeexamples shown and described in this section in connection with thepreferred embodiments and methods. The invention according to itsvarious aspects is particularly pointed out and distinctly claimed inthe attached claims read in view of this specification, and appropriateequivalents.

The embodiments and methods of this invention are useful for opticallyanalyzing test objects, also referred to herein as test pieces, and morepreferably the surfaces of test objects, particularly at highmagnifications or resolutions. Examples of test objects would include,without limitation, semiconductor wafers, semiconductor devices,magnetic disk drives, and the like. The term “surface” as the term isused herein refers in its broadest sense not only to the top or externalsurface or plane, but typically to the subsurface as well. In thesemiconductor field, for example, the surface may include one or morefilm layers, such as dielectric films, metallization, and the like.

In accordance with various aspects of the invention, an apparatus isprovided for analyzing surface characteristics of a test object. Asnoted herein above, such apparatus preferably analyze not only the topsurface or plane, but the subsurface structure as well. The analysispreferably comprises not only height and/or topology of the surface, butsubsurface characteristics as well, for example, such as surface or filmcomposition, e.g., the number of films on the surface, their respectivethicknesses, compositions, refractive index, etc.

In the various preferred embodiments and preferred methodimplementations, or in connection with them, a scanning interferometrysystem is provided for scanning the surface of the test object with anoptical beam and generating an interferometric signal embodyinginformation about the surface of the test object.

Generally in interferometers for optical surface analysis, theilluminating beam is split into a reference beam and a test beam. Thetwo beams are reflected by the reference surface and the test surfacerespectively, recombined to form an interferometric signal, and relayedto a camera sensing the interferograms. The camera signal is digitizedand further analyzed in a computer. During the data acquisition, theoptical path difference (“OPD”) between the reference beam and test beamis varied while a number of camera frames are acquired. From theacquired interferogram intensities in each camera frame, the surfaceheight at test surface points conjugate to the detector elements(pixels) of the camera is calculated, and a two-dimensional surfacetopography map is generated.

A variety of interferometer types are commonly used in interferometricsurface metrology, e.g., Fizeau, Twyman-Green, Mach-Zehnder. (D.Malacara ed., Optical Shop Testing, 2nd Edition, John Wiley, NY 1992.)Specialized microscopic interferometer types include the Mirau,Michelson, and Linnik. (W. Krug, J. Rienitz, G. Schulz, Contributions toInterference Microscopy, Hilger & Watts, London, 1964.) Other than theFizeau-interferometer, all of these interferometer types can be usedwith broadband illumination at an appreciable working distance betweenthe test surface and the interferometer components. This is due to thefact that the OPD between test and reference beam can be adjusted tozero for a non-zero spacing between the test surface and the beamsplitter. The methods and techniques according to the various aspects ofthe invention as disclosed herein below generally can be applied to allinterferometer types in which a broadband or multi-spectral light beamcan be used to obtain an interference signal. The presently preferredembodiments and presently preferred method implementations according tothe various aspects of the invention as described herein below are basedprimarily on the Mirau interference microscope shown in FIG. 1 to betterillustrate these aspects of the invention, although this is notnecessarily intended to be limiting.

The Interferometry System (Hardware) of FIG. 1

To illustrate the principles of the invention according to its variousaspects, a schematic of a scanning interferometry system in the form ofa Mirau-based interference microscope or interferometer 100 according toa presently preferred embodiment of one aspect of the invention is shownin FIG. 1. This interferometer 100 is adapted to analyze the surfacetopology and other characteristics of a test object or piece 10, whichmay comprise a semiconductor wafer, a semiconductor device, or otheritem.

Interferometer 100 includes a light source 1 that projects an opticalbeam 1 a. Light source 1 preferably comprises a multi-spectral lightsource, i.e., comprising multiple frequency components. More preferably,light source 1 provides broadband light, such as white light, emitted ata range of wavelengths. “Broadband” as used herein is used according toits common meaning in the optics and optical engineering field.Typically it is meant to include light having wavelength components ofat least about 100 nanometers (“nm”) range. A preferred embodiment for alight source with broadband light in the visible wavelength range wouldcomprise a halogen lamp. Optical beam 1 a as it is projected from lightsource 1 comprises white light.

Beam 1 a is imaged by lenses 2 and 3 onto an aperture stop 4. The lightpassing through the aperture stop 4 is imaged by lenses 5 and 6 onto astop 7 of a microscope objective 8 after being partially reflected by abeam splitter 9. For the common telecentric microscope objectives, thisobjective stop 7 is in the rear focal plane of the objective, such thatthe light source is imaged to infinity in the object space where testsurface 10 is situated. A field stop 11 is positioned between lenses 5and 6, such that it is imaged to infinity by lens 6, and in turn imagedonto the test surface 10 by the microscope objective 8. The test surface10 is then imaged by the microscope objective 8 and a tube lens 12 ontoa camera 13, with the beam now being partially transmitted by beamsplitter 9. Thus the microscope comprises two complementary imagingtrains, i.e., the test surface imaging, shown in solid lines in FIG. 1,and the pupil imaging (or source imaging), shown in dashed lines in thatsame figure.

A Mirau adapter 14 is placed between objective 8 and test surface 10. Itincludes an interferometric beam splitter 15 and a reference surface 16.The distance between the reference surface 16 and the beam splitter 15is z_(r), and the distance between the test surface 10 and the beamsplitter 15 is z_(t).

In accordance with this aspect of the invention, the system comprisesprocessing means that performs processing on the interferometry signalas described more fully herein below. The processing means preferablybut optionally comprises a processor, for example, such as amicrocontroller or the processor of a general purpose computer,programmed and/or otherwise configured to perform the tasks andfunctions as described more fully herein below. The processor and/or theprocessing means may comprise more than one actual processing devices,such as multiple central processing units (“CPU”) in a distributedprocessing environment, or wherein the signal is processed in more thanone piece of equipment. Indeed, the processor itself thus configuredcomprises an aspect of the invention as well. The processor also maycomprise peripheral devices, for example, such as a keyboard, mouse,track ball, and the like. As implemented in the presently preferredembodiments and method implementations, the processing means comprises aprocessor in the form of a general purpose computer 17, which in turncomprises a processing unit 17 a such as a tower containing amotherboard or similar circuitry including an arithmetic logic unit,random access memory, etc. Processor 17 is operatively (i.e., directlyor indirectly) coupled to camera 13 to receive the interferometrysignal, which is digitized, for example, by camera 13, processor 17 orsome combination of these.

The system also preferably comprises storage means for storing theinterferometry signal, transformations of it, data from it, a pluralityof models corresponding respectively to a plurality of surfaceconditions of the test object, and/or results of comparisons of themodels with the interferometry signal or representations of it, and/orthe program or programs used to generate these quantities. The storagemeans may comprise any device or group of devices capable of performingthe storage functions as described herein. Storage devices compatiblewith processor 17 include, for example, a hard drive, a disk drive, andsimilar commercially available digital storage media. The storagedevices comprise or are adapted to operate with a machine readablestorage medium, such as a magnetic or optical storage medium. Examplesof such storage media include solid state memory, magnetic hard drives,diskettes, compact disks (“CD”), DVDs, jump drives, optical storagedisks, and the like. In system 100, the storage means comprises a massstorage device 17 b, such as a hard drive. Storage device 17 b alsopreferably comprises a removable storage device, such as a disk drive,jump drive, or the like.

The system also may comprise display means for displaying certainfunctions, as are more specifically described herein. The display meansmay comprise any display device capable of providing the displayfunctions as described herein, preferably in cooperation with theprocessor. As implemented in the presently preferred embodiments andmethod implementations, the display means comprises a computer monitor17 c operatively (i.e., directly or indirectly) coupled to processor 17to display the data and functions as described more fully herein below.

In accordance with presently preferred method implementations of theinvention, the scanning interferometry system is used to acquire datafrom the surface of the test object. In so doing, light source 1 is usedto scan a surface location of test object 10 as more fully describedherein. The scan may occur at a single surface location, or morepreferably, the scan occurs sequentially at one surface location, thenanother, and another until the desired portions of the surface have beenscanned. This may or may not include the entire surface, depending onthe application, the desired amount of testing, etc.

In scanning the surface location of the test object, the illuminatingbeam 1 a is split into a reference beam 1 r and the test beam it by theinterferometric beam splitter 15. The reference beam 1 r is the beamportion that is directed toward the reference surface 16, where it isreflected back to beam splitter 15. The test beam 1 t is the beamportion that is directed to the test surface 10, where it is reflectedto beam splitter 15. At beam splitter 15, both beams recombine, andcontinue propagating through the objective 8, beam splitter 9, and tubelens 12 to camera 13 where they form an interference pattern. At eachdetector element (pixel) of the camera 13 this interference pattern istransformed in camera 13 and/or its associated circuitry into aninterferometry signal. This interferometry signal comprises or embodiesa time domain signal, which can be viewed as an intensity versus timerepresentation of the interferometry signal.

It is preferred in connection with the presently preferred embodimentsand methods that the scanning of the surface comprise not only varyingthe OPD in a direction substantially normal to the test surface (hereassumed to be “vertical” or in the “z” direction), but also in dividingthis vertical scan into a plurality of sequential segments, preferablyequal in time and in length. This causes the interferometry signal, aswell as the corresponding time domain and frequency domainrepresentations of it, to be divided into a plurality oftime-differentiated segments, wherein the time domain segments and thefrequency domain segments correspond to one another in time.

During data acquisition and for a given spot or location on the surfaceof test piece 10, the OPD between the test beam it and reference beam 1r is varied while the frames from camera 13 are acquired by processingcomputer 17 for processing. The OPD variation optionally but preferablyis controlled by computer 17 to affect uniform steps in z_(t) betweencamera frames. The test piece is supported by a piezoelectric transducer(“PZT”) actuator 18 which elongates along the optical axis 19 accordingto a control voltage from computer 17 or an intermediate amplifierproviding the required high-voltage signal for the PZT actuator.Alternatively, objective 8 together with the Mirau adapter 14 may beattached to a PZT actuator controlled by computer 17. For otherinterferometer configurations, the PZT actuator 18 can be placeddifferently to utilize the different possibilities of the differentoptical configurations. For example, it can be placed to move the testsurface or the reference surface in a Michelson or Linnikinterferometer, or the reference objective together with the referencesurface in case of a Linnik interferometer.

The field stop 11 serves to reduce stray light in the system 100 bylimiting the extent of the optical beam to just fill the detected imagefield of camera 13. The aperture stop 4 limits the angular extent of theillumination of the test surface 10 if its image in the plane ofobjective stop 7 is smaller than the objective stop itself. This can beseen in more detail in FIG. 2, wherein the Mirau adapter 14 is omittedfor clarity and ease of illustration. The re-imaged source point 20 onthe optical axis 19 of objective 8 is the source of a diverging wavethat is transformed into a plane wave by objective 8 before it impingeson test surface 10 at normal incidence. The re-imaged source point 21 inthe objective stop 7 also is the source of a diverging wave, which againis transformed to a plane wave by objective 8, now however impinging onthe test surface 10 at oblique incidence. Like the beam at normalincidence, the oblique beam is reflected by test surface 10, passesthrough the objective 8, and travels on to camera 13. The further theoff-axis source point 21 is from the optical axis 19, the larger theangle of incidence of the oblique beam. The exact relationship is givenby the sine-condition:sin θ=r/f  (1)where θ is the angle of incidence, r is the distance of source point 21from the optical axis 19, and f is the focal length of the objective 8.This equation is also used for the definition of the numerical aperture(“NA”) of objective 8:NA=n·sin(θ_(o))  (2)where n is the refractive index of the medium in object space and θ_(o)is the maximum angle of incidence given by the radius r_(o) of theobjective stop 7.

As noted, light source 1 in the presently preferred embodiments andpresently preferred method implementations is a broadband light source.Thus the interferogram intensity detected at each pixel of the camera 13as function of the test surface position z_(t) is given by asuperposition of beams of different wavelength and different angle ofincidence in object space, (G. Kino, S. Chim, Mirau correlationmicroscope, Applied Optics 29, 3775-3783): $\begin{matrix}{{I\left( z_{t} \right)} = {{const} + {{Re}\left\{ {\int_{k_{\min}}^{k_{\max}}{{G(k)}{S(k)}{\int_{0}^{\theta_{\max}}{{L(\theta)}{\rho_{t}\left( {k,\theta} \right)}{\rho_{r}^{*}\left( {k,\theta} \right)}{\mathbb{e}}^{{ik}\quad 2{({z_{t} - z_{r}})}{\cos{(\theta)}}}{\sin(\theta)}{\cos(\theta)}{\mathbb{d}\theta}{\mathbb{d}k}}}}} \right\}}}} & (3)\end{matrix}$where k is the wavenumber given by 2π/λ, and λ is the wavelength, G(k)is the spectral power density of the light source 1 between k_(min) andk_(max), S(k) describes the spectral transmittance of the common-pathpart of the interferometer system 100 from the light source 1 to thecamera 13 as well as the spectral responsivity of camera 13, L(θ)describes the light intensity at angle θ, ρ_(t)(k,θ) is the complexspectral reflection coefficient of the test surface 10 at wavenumber kand angle θ, ρ_(r)(k,θ) is the complex spectral reflection coefficientof the reference optics at wavenumber k and angle θ, including theamplitude and phase effects of the reference surface 16 andinterferometric beam splitter 15, i is the imaginary unit, Re signifiesthe real part of the complex number in the braces { }, and the asteriskdenotes the complex conjugate.

The above equation (3) assumes a rotationally uniform illumination ofthe objective stop 7 where the wavelength effects can be factorized fromthe spatial effects. Furthermore, all polarization effects areneglected. To account for polarization, equation (3) can be extended,e.g., using the coherency-matrix formalism. (M. Born, E. Wolf,Principles of Optics, 7th edition, Cambridge University Press,Cambridge, 1999.)

In optimizing the performance of interferometer 100, and in thepreferred method implementations as described more fully herein below,it is usually desirable to calibrate the interferometer and to cancelout contributions to the interferometry signal outputted from theinterferometer that are attributable to the machine itself Equation (3)shows that the properties of the reference optics of the Mirauinterferometer 100 are mixed with the properties of the test surface 10under the θ-integral. Thus, a rigorous calibration and elimination ofthe reference optics properties requires knowledge at all wavenumbersand all angles. In order to simplify the tool calibration and also makethe suppression of polarization effects in the interferometer opticsvalid, the range of illumination angles can be reduced such that theθ-variation in equation (3) can be neglected. This may be achieved byusing an objective 8 with a sufficiently small numerical aperture(“NA”). However, the imaging resolution of the test surface 10 is lowerfor smaller NA. To maintain a higher imaging resolution, an objective 8with higher NA can be used while the aperture stop 4 is reduced in size,and the illuminating NA is smaller than the imaging NA. For typicalcases of test surfaces such as CMP products, an illuminating NA of lessthan 0.15 is suitable [00]. For data acquisition, the z-position of thetest surface 10 is scanned at a given location on the surface whileframes of the interferogram intensities are acquired by camera 13. Forfurther processing, the varying part of the intensity signal is ofinterest. Switching to complex notation, the varying intensity part ofequation (3) is obtained as function of scan position z with thesimplification of low NA illumination:i(z)=∫F(k)ρ_(t)(k)ρ_(r)*(k)e ^(2ik(z+z) ^(i) ⁾ dk  (4)where F(k)=G(k)S(k) is a real function containing the spectral radianceof the light source 1, the camera responsivity and common-path opticalproperties of the interferometer optics in a suitable normalization, andz_(t) is now the test surface position for z=0 with respect to z_(r),i.e., it replaces z_(t)−z_(r) in equation (3). Combining allinterferometer parameters into one complex tool function H(k) with:H(k)=F(k)ρ_(r)*(k)  (5)we obtain for equation (4):i(z)=∫H(k)ρ_(t)(k)e ^(2ikz) e ^(2ikz) dk  (6)Fourier transforming this intensity signal yields the complex spectralreflection coefficient of the test surface, multiplied with the toolfunction H(k) and a phase factor, (B. Lee, T. Strand, Profilometry witha Coherence Scanning Microscope, Applied Optics 29, 3784-3788, 1990):S(k)=FT{i(z)}=H(k)ρ_(t)(k)e ^(2ikz) ^(i)   (7)In order to completely characterize or calibrate the interferometer tool100, H(k) needs to be determined. To do this, a known surface can bemeasured with the interferometer, i.e., a calibration surface with aknown complex spectral reflection coefficient r_(c)(k) is placed as testsurface 10 in the interferometer 100. Fringe signal i_(c)(k) is obtainedfrom a measurement of this calibration surface:i _(c)(z)=∫H(k)ρ_(c)(k)e ^(2ikz) ^(c) e ^(2ikz) dk  (8)where z_(c) is the position of the calibration surface. To explicitlyobtain H(k), the intensity signal i_(c)(z) is Fourier transformed andthe known spectral reflection coefficient ρ_(c)(k) is eliminated:H(k)e ^(2ikz) ^(c) =FT{i _(c)(z)}/ρ_(c)(k)  (9)Note that the complex function H(k) is only known to within a linearphase term depending on z_(c). Thus z_(c) can be considered the newz-coordinate origin to which all test surface measurements arereferenced when the tool 100 is calibrated. The calibration is thissimple only with reduced illumination angles. Without this anglerestriction, the calibration of the interferometer requires calibrationmeasurements with different angles of incidence in test surface spacesuch that ρ_(r)(k,θ) is explicitly obtained.

In the spectral domain, the tool function now can be eliminated from themeasurement:ρ_(t)′(k)=ρ_(t)(k)e ^(2ikz) ^(t) =S(k)/H(k)  (10)where z_(c) has been set to 0 as the new z-coordinate origin. Equation(10) can be executed wherever the tool function H(k) is unequal to zero,or for good noise performance where the tool spectrum has sizeableamplitudes.

In accordance with various aspects of the invention, the interferometrysignal and/or the data acquired from it are transformed into ann-dimensional function comprising an independent frequency variable andan independent time variable. According to another related aspect of theinvention, the interferometry signal and/or data from it are transformedto an n-dimensional function having an independent scale variable and anindependent time variable. The n-dimensional function preferably is2-dimensional, although others are possible. According to yet anotheraspect of the invention, a window function is applied to theinterferometry signal to obtain a windowed interferometry signal,obtaining a frequency domain signal from the windowed interferometrysignal, and constructing a multi-domain function from the frequencydomain signal and the time domain signal. The multi-domain functionpreferably is a 2-domain function although, again, others are possible.

Although not wishing to be limited to any particular theory ofoperation, it is noteworthy that, in the spectral domain, ρ′_(t)(k)contains the information about the test surface properties as well asthe test surface position, i.e., test surface height. The spectralcoefficient can provide information about the composition of thesurface, for example, such as the refractive index or indices of a filmlayer or layers comprising the surface. If the spectral reflectioncoefficient of the test surface is known, it can be eliminated fromρ_(t)′ and one can obtain z_(t) in a number of ways. (See, e.g., P.DeGroot, X. C. DeLega, J. Kramer and M. Turzhitsky, Determination ofFringe Order in White Light Interference Microscopy, Applied Optics 41,4571-4578, 2002). However, if the spectral reflection coefficient of thetest surface is unknown, the surface topography cannot be independentlyobtained.

Depending on the material and structure of the test surface 10, threedifferent cases mainly can be considered for the spectral reflectioncoefficient.

In the first case the test surface is the top surface of a bulkmaterial. For example, it may be the top surface of a thick dielectricmaterial where the bottom surface does not contribute to theinterference signal, or it may be the top surface of an opaque metal.The spectral reflection coefficient is given by the refractive index ofthe material. (See, e.g., Born, above). $\begin{matrix}{{\rho_{t}(k)} = \frac{{N_{0}(k)} - {N_{1}(k)}}{{N_{0}(k)} + {N_{1}(k)}}} & (11)\end{matrix}$where N₀(k) is the refractive index of the object space (real), andN₁(k) is the refractive index of the test surface material. If the testsurface consists of a dielectric material, N₁ is real and the phaseshift on reflection is either 0 or π radians. If the test surfaceconsists of a metal with some absorption, N₁ is complex with:N ₁(k)=n ₁(k)−iκ ₁(k)  (12)where n₁(k) is the real refractive index and κ₁(k) is the extinctioncoefficient.

In the second case the test surface is the top surface of a single ormulti-layer stack of thin films on a substrate. The complex reflectioncoefficient depends now on the refractive index and the thickness ofeach layer as well as the substrate. It can be calculated by knownmatrix techniques. (See, e.g., H. A. McLeod, Thin-film Optical Filters,3rd Edition, Institute of Physics Publishing, Bristol, 2001.) Additionalfactors, such as surface roughness and interface, also can be accountedfor by modeling.

In the third case, the test surface is a micro-structured surface. Thinmetal lines, for example, may be embedded in a surrounding dielectriclayered material on a wafer surface after CMP. The complex reflectioncoefficient can be calculated by vector diffraction methods such asRigorous Coupled Wave Analysis (“RCWA”). (See, e.g., M. G. Moharam, E.B. Grann, D. A. Pommet, Formulation of Stable and EfficientImplementation of the Rigorous Coupled Wave Analysis of Binary Gratings,J. Opt. Soc. Am. A12, 1068-1076, 1995; and M. G. Moharam, D. A. Pommet,E. B. Grann, Formulation of Stable Implementation of the RigorousCoupled-Wave Analysis for Surface Relief Gratings; EnhancedTransmittance Matrix Approach, J. Opt. Soc. Am. A12, 1077-1086, 1995.)

Only in the simple first case is it possible to directly invert thespectral reflection coefficient to obtain the refractive index of thematerial. In the two other cases it is not possible to use a directinversion to obtain detailed surface parameters. To determine the layermaterials and thickness, and/or micro-structure parameters, a fittingtechnique can be applied that compares the measured spectral reflectioncoefficients with calculated model reflection coefficients and selectsthe model with the best match. This numerical technique has been appliedin other measurement instruments such as a spectrophotometer,ellipsometer, or scatterometer.

In the cases described herein above, it has been assumed that allmaterials are isotropic and there is no dependence on the plane ofpolarization at normal incidence. This is usually not true formicro-structured surfaces that likely will have preferential directionsand different reflection coefficients for different polarizations. Ifthere is any anisotropy in the surface response, one or moreinterferometric measurements can be carried out with polarized light ofdifferent states of polarization, providing additional information forthe fitting process.

The fitting process for determining the detailed structure of the testsurface experiences two main problems. The first problem is thepotential correlation of the z-position of the test surface with thesurface model parameters such as layer thickness or material index.Techniques exist to overcome this problem, for example, by basing thefit on the amplitude of the spectral reflection coefficient, or byeliminating the spectral phase slope before the fit. The second problemarises when there are a large number of fit parameters, leading to alack of convergence, which in turn can result in large uncertainties.This problem may occur in the case of multi-layers with many freeparameters such as layer thickness or material. This problem of too manyfit parameters is not overcome by the fitting directly in the z-domain,as described in U.S. Patent Publication No. US 2004/0189999 A1, De Grootet al.

To overcome these difficulties, a mixed-domain signal analysis techniqueis used in the present embodiment and present method implementations.This optionally but preferably may be combined with an optimized lightsource to allow for separating signal components from lower layers in afilm stack. Although again not wishing to be limited to any particulartheory, this aspect of the invention takes advantage of the fact thatthe response from multilayer-stacks can be accurately modeled as asuperposition of a beam reflected at the top surface with an infinitenumber of beams experiencing an increasing number of internalreflections within the multilayer stack before exiting. Hence thecomplex reflection coefficient may be written: $\begin{matrix}{{\rho_{t}(k)} = {\sum\limits_{q = 0}^{\infty}{{A_{q}(k)}{\mathbb{e}}^{{i\alpha}_{q}k}}}} & (13)\end{matrix}$where A_(q)(k) is the complex amplitude of the qth beam containing thenon-linear phase terms from the reflections as well as from thepropagation through layers with dispersion, and α_(q) is the effectivez-shift of the qth beam with respect to the top surface. The index q=0signifies the beam reflected at the top surface. With equation (6) wecan take the Fourier transform of the time signal and obtain:$\begin{matrix}{{i(z)} = {\sum\limits_{q = 0}^{\infty}{e_{q}\left( {z - \alpha_{q}} \right)}}} & (14)\end{matrix}$where the qth component e_(q) is shifted by α_(q) and given by:e _(q)(z)=h(z)*α_(q)(z)  (15)where h(z) is tool z-response function given by the Fourier transform ofthe tool function H(k), a_(q)(z) is the Fourier transform of thespectral amplitude variation of the qth beam, and * represents theconvolution operator.

As noted herein above, preferred system embodiments and methods comprisea vertical scanning aspect in which the vertical scan is divided into aplurality of time segments. Accordingly, the scanning interferometrysystem comprises means for vertically scanning a position on the surfaceof the test object, and the processor comprises means for dividing theinterferometry signal into a plurality of time-differentiated segments.The means for dividing the interferometry signal into a plurality oftime-differentiated segments preferably spaces the time-differentiatedsegments so that each of the time differentiated segments has an equaltime duration, and preferably an equal length as well. The means fordividing the interferometry signal into a plurality oftime-differentiated segments preferably divides the plurality of timedifferentiated segments so that the time differentiated segments overlapone another.

The means for vertically scanning a position on the surface of the testobject may take a number of forms, provided it produces the appropriatescanning function, e.g., the appropriate relative movement to vary theOPD between the reference surface and the test surface. Either surfacemay move, for example, or both surfaces may move. In the presentlypreferred embodiments and method implementations, this means comprisesPZT actuator 18.

The means for dividing the interferometry signal into a plurality oftime-differentiated segments also may take a number of forms, butpreferably comprises a processor, e.g., processor 17, alone or incombination with circuitry in camera 13. These time-differentiatedsegments preferably comprise camera frames interposed sequentially ascamera 13 records the images of light beam 1 b from the interferometrysystem 100.

Typically many fit parameters in a multilayer stack are of concern onlywhen the top layers are transparent, i.e., dielectric. In this case, theA_(q)(k) are close to constant with a constant phase shift, and thea_(q)(z) are, to good approximation, a weighted delta-function. Thus, atleast for the top dielectric layers of a multilayer film stack theintensity response is a complex superposition of tool z-responsefunctions shifted by the effective shifts α_(q). For a dielectric toplayer, the shift α₀ is zero. The further the beam penetrates into thefilm stack, and the more internal reflections it experiences, the moreit is delayed with respect to the beam reflected at the top surface, andthe larger α_(q) will be.

Taking the z-shift information in the z-domain into account at the sametime as the spectral information by using a mixed-domain approach allowsfor optimized fitting of the test surface parameters as well as thesurface topography.

The details of the data processing are described below. For the purposeof that discussion, the z-coordinate and the time coordinate are usedsynonymously, since the actual scanning in the interferometer occurs intime.

For many applications, classical signal analysis strategies are based ontime and frequency. Time-domain methods are adequate for tasks such asedge detection, elementary segmentation, and texture analysis problems.But in many situations, the inherent periodicity within a signal pushestoward a decomposition of the signal according to its frequency content,that is Frequency-domain analysis. It is a useful tool for discoveringthe sinusoidal behavior of a signal. The spectral analysis methods, suchas the Fourier transform, permit the solution of some problems thatconfound time-domain techniques. However, they suffer signalinterpretation difficulties when oscillations of interest exist onlywithin a limited time interval since it is an inherently globalapproach.

The input signal directly obtained from a surface with a multilayer filmstack by the scanning interferometers such as described above is anon-stationary signal, that is, it is a signal whose properties evolvein time. If it is transformed into the frequency domain by Fouriertransform, for example, as proposed by Kim et al. and De Groot et al.(mentioned above), any local information change in time is spread outover the whole frequency domain. Therefore, an analysis adapted tonon-stationary signals requires more than a Fourier Transform.

In accordance with this aspect of the invention, presently preferredembodiments and preferred method implementations mix time-domainapproaches with the frequency- and scale-domain approaches. Bothcombinations provide non-stationary signal analysis. Preferred apparatusand methods decompose a one-dimensional (“1-D”) time signal into atwo-dimensional (“2-D”) mixed-domain function that provides a better wayfor differentiating between desired information and unwantedinformation. This is valuable in applications where it is necessary ordesirable to identify information localized in time, for example, asexperienced in the applications disclosed herein.

Two processing techniques are used in accordance with these presentlypreferred embodiments and methods. One is a time-frequency analysis, andthe other is a time-scale analysis.

In accordance with a related aspect of the invention, a window functionis applied to the interferometry signal to obtain a “windowed”interferometry signal, preferably in the processor, and the surface ofthe test object is characterized using the windowed interferometrysignal. This preferably comprises applying multiple windows to theinterferometry signal, and more preferably applying multiple windowssequentially over time. Similarly, the invention according to relatedaspects comprises transforming the data acquired from the interferometrysystem to the n-dimensional function by applying a window function tothe interferometry signal, preferably using multiple windows and morepreferably using multiple windows sequentially over time.

As implemented in presently preferred apparatus and methods, thetime-frequency analysis slides a time window over the input signal toproduce frequencies depending on time. The result from thetime-frequency analysis is a 2-D function, called a spectrogram. Thetime-scale analysis scales the time window width at differentfrequencies to produce scales depending on time. It provides analternative to time-frequency analysis, which uses a single analysiswindow. The result from the time-scale analysis is an n-dimensionalfunction, preferably a 2-dimensional (“2-D”) function, called ascalogram. Both techniques result in an n-dimensional transformed signalhaving an independent frequency or scale variable and an independenttime variable. Thus, the presently preferred embodiments and methodsaccording to this aspect of the invention capture both the frequency orscale components of the input signal and their time locality. Thesespectrograms and scalograms provide much more rich and detailedinformation related to the properties of the test surface as well as themeasuring system than the 1-D spectrum from the Fourier transform.

A typical temporal intensity distribution, obtainable directly fromscanning interferometer 100 for a test surface 10 with multipletransparent thin films, is depicted in FIG. 6. This is a time signalbecause its data are acquired sequentially as the vertical or z-positionchanges in time. The time-frequency analysis is selected to process thetemporal intensity distribution i(z), or in general the non-stationarytime signal x(z). The general form for time-frequency analysis isdefined by: $\begin{matrix}{{X\left( {b,\omega} \right)} = {\int_{- \infty}^{\infty}{{x(z)}\frac{1}{\sqrt{a}}{g\left( \frac{z - b}{a} \right)}{\mathbb{e}}^{{- {j\omega}}\quad z}{{\mathbb{d}z}.}}}} & (17)\end{matrix}$where g(z) is a window function, a is a fixed scale factor, w is theangular frequency, and b is the location of the window in time or thetemporal position z. By selecting the Gaussian window for g(z) and a=1,a Gabor transform is obtained. In this case the result for thetime-frequency analysis of the time signal of FIG. 6 is a spectrogramwhose amplitude is depicted in FIG. 7. Amplitude in FIG. 7 displays asbrightness, the lightest areas being greatest amplitude. Note that they-axis is the angular frequency w. Both phase and amplitude values ofthe spectrogram are directly related to the position z.

There are many window functions g(z) other than the Gabor transform,such as B-spline, Hann, Hamming, Blackman, Harris-Nutall windows, etc.The window function, in fact, can be any function that decreases withdistance from its center. The window width can be an important parameterto be determined in every application. To extract more localized timeinformation, a shorter window width can be selected. The window widthchosen in this application depends on the width of the tool z-responsefunction h(z), which is determined by the tool function H(k). Thenarrower h(z), the smaller is the window width and the better is timedomain resolution. This is because any data outside the window does nothave any effects on determining surface properties and height inside thewindow. Thus, the shape and width of the tool function, and hence thelight spectrum, can play an important role in the instrument.

Having generated the transformed n-dimensional function, it is thencompared with a set of models to determine a best match from thecomparison. Similarly, having generated the multi-domain function, italso may be compared with a set of models to determine a best match fromthe comparison.

The n-dimensional function and/or multi-domain function generated fromthe interferometry signal is compared with a set of comparable models,preferably functions similar to those of the n-dimensional functionand/or the multi-domain function from the interferometry signal,respectively, that represent various surfaces and surface conditions orcharacteristics hypothesized to be present with the test surface. Thesemodels preferably reflect different surface properties, such as films,film thicknesses, refractive indices, time position, line width, etc.that may be present at the test surface. The comparison may take theform of parameters of the functions and/or models as well, oralternatively. In the presently preferred embodiments and methods, eachfunction from each sample of the interferometry signal, i.e., eachinterferogram obtained for the test object, is compared with each modelto ascertain a best match from among the models with the function fromthe test object. From this best match, the surface structure,characteristics, etc. that constitute or correspond with the bestmatching model are selected as the surface structure, characteristics,etc. of the test object under analysis. The surface properties aredetermined from the best match.

Preferably, the analysis and comparison comprises applying anappropriate weighting function to the n-dimensional functions and/or themulti-dimensional function. This can produce a robust and accuratecomparison between the functions and the models, but also can alleviateor eliminate unwanted information that otherwise would be used in thecomparison.

The comparison and selection of a best match from among the models maybe carried out in a number of ways. Preferably it is carried out foreach spot or location on the test object surface that is to be tested,so that a model match is obtained at each such location. Comparisonsand/or correlations also may be made from surface spot or location tospot or location, across the surface or a region of it.

The comparison itself also may take a number of forms. It may, forexample, involve selecting one or more parameters from the functions forcomparison, for example, such as an intensity measure and a phase shiftmeasure for a particular interferogram. The comparison also may usevarious techniques to optimize the selection of the model based onpre-determined criteria.

Using equation (6), a set of time signals can be generated with a knownlight spectrum, a constant z-step of the scan, and with differentproperties of the test surface, such as the film thickness andrefractive index of materials. In the presently preferred embodimentsand methods, each of these signals is transformed by equation (17) to aspectrogram that is compared with the one obtained from the datadirectly from the instrument. A 2-dimensional weighting function W(b,w)with larger weights for the more interesting regions and smaller weightsor zeros for other areas in spectrograms is used in the comparison. Inother words, the information in different areas of the spectrogram maybe emphasized differently in the matching process. Thus, spectralcomponents in frequency ranges and time ranges of interest can becompared. Once the best match is determined, the surface properties usedfor the best-fitting model can be selected as the desired properties.The time position of the input signal (the interferometry signal)relative to the model signal, i.e., the relative surface height, can bedetermined, for example, from the time position for best amplitudematching together with the phase difference between their spectrograms.

The time-frequency analysis approach described herein presents acomplete structural description of a signal. As noted, it preferablycomprises trimming the source signal x(z) with a decaying time windowfunction. This technique time-limits, or windows, a signal beforecalculating its spectrum. Windowing furnishes better estimates of asignal's spectrum, for example, because it restricts the signal valuesto those over which the relevant oscillatory waveform features shouldappear. Windowing inherently provides a true local spectrum that is freefrom the influence of data outside the window. In modeling, this impliesthat the properties of those films and of the substrate far away fromthe top surface do not need to be known in profiling the top surface orin finding the top film parameters. This inherent property oftime-frequency analysis gives embodiments and methods according to theseaspects of the invention a great advantage over other methods, forexample, such as those based on Fourier transforms.

In order to increase the processing speed and robustness of finding thedesired surface properties with the spectrogram comparison, a set oftime signals can be generated that matches the time phase at a fixedlocation of the input signal, such as the phase at the maximum envelopeposition, or the phase at another time position of interest. This set oftime signals can be obtained by equation (6) with changing timepositions z. Thus, spectrograms can be compared without relative timeposition changes. Similarly, the desired surface properties can bedetermined once the best matching spectrogram is known. The timeposition z for generating the time signal for the best matchingspectrogram normally is the relative surface height. There are othertime-frequency analysis techniques available, such as Wigner-Ville andZak transforms, (see, e.g., R. D. Allen, D. W. Mills, Signal Analysis,IEEE Press, 2004), which have forms different from Equation (17). Theyalso may be used to transform the time signal to the time-frequencydomain.

In addition to time-frequency analysis, the time-scale analysis orwavelet transform (WT) can be used for the analysis of non-stationarysignals. In contrast with time-frequency analysis, which uses a singlefixed window, the WT uses short windows at high frequencies and longwindows at low frequencies. Thus, it has a varying time-frequencyresolution that results in more effective handling of signals withtransients and components whose pitch changes more rapidly thantime-frequency analysis. The time-scale or WT thus may be especiallyvaluable in the analysis of an input signal from a surface that has verythin films and any sudden signal changes.

The wavelet transform uses a signal scale variable instead of afrequency variable in its transform. The general form of the wavelettransform of a signal is given by the coefficient W(a,b):$\begin{matrix}{{W\left( {a,b} \right)} = {\int_{- \infty}^{\infty}{{x(z)}{\psi_{a,b}^{*}(z)}{{\mathbb{d}z}.}}}} & (18)\end{matrix}$where the superscript * denotes the complex conjugate, and a completeset of daughter wavelets is generated from the mother wavelet ψ(z) bydilations and translations: $\begin{matrix}{{\psi_{a,b}(z)} = {\frac{1}{\sqrt{a}}{\psi\left( \frac{z - b}{a} \right)}}} & (19)\end{matrix}$where a is the scale factor, b is the translation factor, and 1/√{squareroot over (a)} is the normalization factor.

There are various wavelets available, (see, e.g., A. D. Poularikas, TheHandbook of Formulas and Tables for Signal Processing, CRC Press LLC,1999). By using Morlet wavelet defined as: $\begin{matrix}{{\psi_{a,b}(z)} = {\frac{1}{\sqrt{a}}{g\left( \frac{z - b}{a} \right)}{\mathbb{e}}^{{j\omega}_{c}{z/a}}}} & (20)\end{matrix}$where g(z) is the Gaussian window and ω_(c) is the center of angularfrequency, the result from the wavelet transform of the time signal,shown in FIG. 6, is a scalogram whose amplitude is depicted in FIG. 8.Unlike the spectrogram, its y-axis is the scale factor a. All theprocess steps in time-scale analysis for this application are the sameas in the time-frequency analysis except using the scalogram instead ofthe spectrogram.

After the data acquisition and the extended mixed-domain processing asdescribed herein, depending on the test surface, two-dimensional orn-dimensional maps of top-surface topography, film-thickness profiles,material distribution, and micro-structure parameters such as line widthor line thickness can be generated. These maps can be used as the finalresult of the measurement. In addition, a host of information can beextracted from these maps. For example, in the case of CMP processcontrol, special parameters of interest may be extracted, such asdishing, erosion, and top layer film thickness. Furthermore, thethickness of opaque metal structures can be inferred from the topsurface topography in combination with the thickness of dielectric filmsadjacent to the copper structure.

In accordance with another aspect of the invention, apparatus andmethods are provided for modifying the spectral composition of the lightbeam so that it facilitates the desired task of obtaining informationabout the test object surface. One of course may select the light sourceusing the spectral composition of the light beam as a selectioncriteria, e.g., so that it comprises or consists essentially of thosespectral components that provide beneficial results for the overallprocess. One also may modify the light beam produced by the light sourceso that the light beam has the desired spectral composition, forexample, using a filter or other optics to condition the beam. Inaddition or alternatively, one may use signal processing techniques tomodify the interferometry signal, for example, at the camera, in theprocessor, or both, to obtain the desired spectral composition.

In order to support this data processing procedure in the presentlypreferred embodiments and method implementations, the tool function H(k)of the interferometer is optimized for separation of the signals fromthe lower layers of multi-layer stacks from the top surface signal. Withthis separation, a successful fit of layer parameters of the upperportion of the multilayer stack can be carried out without overloadingthe fit procedure with too many fit parameters corresponding to lowerlayers, or converging to the wrong model due to insufficient informationabout the lower layers.

According to equation (5), the tool function H(k) is affected by thespectral radiance of the light source 1, the optical properties of thecommon-path interferometer optics, the optical properties of theinterferometric beam splitter 15 and reference surface 16, and thespectral responsivity of camera 13. For a given interferometer, thelight spectrum going into the interferometer can be affected to changeH(k).

As an example to illustrate this aspect of the invention, an opticalfilter such as an interference filter 22 with a desired spectraltransmittance function can be placed between lenses 2 and 3 of theinterferometer to modify light beam 1 a, as shown in FIG. 1.

In another embodiment, the light source 1, lenses 2 and 3, and thespectral filter 22 may be placed remotely in a separate unit where theoutput is coupled to the interferometer by an optical fiber or fiberbundle. In this case the output end of the fiber is placed directly inthe plane of the aperture stop 4.

In yet another embodiment, the light source 1 can be replaced by theoutput aperture of a spectral shaping device. FIG. 3 shows a schematicof such a device. The broadband light from a light source 23 withappropriate aperture is collimated by lens 24, dispersed by prism 25,and focused by lens 26 onto the spectral dispersion plane 27. In thespectral dispersion plane 27 the spectral content of the beam ismodified by a filter 28 as described below. The beam then propagatesfurther to lens 29, where it is collimated again. Prism 30 reverses thespectral dispersion, such that lens 31 focuses the light of allwavelengths onto the output aperture of the spectral shaping device ataperture 32.

Another embodiment of a spectral shaping device according to this aspectof the invention is shown in FIG. 4. There light from source 23 iscollimated by lens 24 and dispersed by prism 25. Prism 25 now functionsas a Littrow-prism that retro-reflects the dispersed beam. The reflectedbeam is focused by lens 24 onto the spectral dispersion plane 27, whereit is filtered by filter 28. From the filtering plane, the lightpropagates to lens 29, and further to prism 30 which reverses again thespectral dispersion, and is focused by lens 29 on the output aperture32.

Note that in FIG. 4 the direction of dispersion in plane 27 is normal tothe drawing, whereas in FIG. 3 the direction of dispersion is in thedrawing plane.

Yet another embodiment is shown in FIG. 5, which differs from FIG. 4 inthat the spectral filtering in the dispersion plane 27 is combined witha reflection.

Plane gratings also can be used as a dispersive element instead of theprisms in FIGS. 4 and 5. Concave gratings combining the dispersion withthe focusing power of lenses 24 and 29 also could be used instead of thelens-prism combinations. Furthermore, the reflection in the dispersionplane 27 can be arranged such that the filtered light is directed backto the light source. However, this requires a beam splitter to deflectthe filtered light resulting in a significant loss of light, andnormally is not preferred.

In alternative embodiments, the spectral shaping device according toFIGS. 3, 4 or 5 is not directly placed on the interferometer setup, butinstead is situated remotely and coupled to the interferometer, forexample, with an optical fiber or fiber bundle. In this case, the inputend of the fiber is placed in the output plane 32 of the spectralshaping device, and the output end of the fiber is replaces the lightsource 1 of the interferometer. Alternatively, the output end of thefiber can be placed directly in the plane of the aperture stop 4.

Also the illumination of the spectral shaping device may occur throughan optical fiber or fiber bundle.

In the dispersion plane 27, light of different wavenumbers k is spreadout along the direction of dispersion. Calling the direction ofdispersion the x-direction, the spectrum can be modified by changing theeffective transmittance T(x) of the filter 28 for FIG. 3 and FIG. 4. Fora fixed filter function, an aperture can be used in plane 27 whose widthnormal to the x-direction varies such that more or less light of eachwavenumber is blocked. For FIG. 5, the effective reflectance R(x) of thefilter 28 is controlled by combining the aperture with a mirror in closeproximity.

In accordance with a related aspect of the invention, the spectralshaping device may comprise a programmable filter, which in thepresently preferred embodiments and methods may comprise programmablespectral filter 28. Programmable filter 28 is operatively coupled to andcontrolled by computer 17. For the transmissive case a liquid crystalbased spatial light modulator is used, whereas for the reflective case aLiquid-Crystal-On-Silicon (“LCOS”) light modulator or Digital MirrorDevice (“DMD”) can be used. These modulators can be operated in anon-off mode for each pixel, thus emulating the aperture described above.A programmable filter allows for optimization of the tool function H(k)based on calibration measurements of the actual interferometer. Thus,part-to-part variations between different instances of theinterferometer can be accounted for, as well as aging of opticalcomponents and the light source, etc. Furthermore, the optimized toolfunction can be maintained even when the interferometer configuration ischanged, e.g., by placing a polarizer or attenuator in the beam path.

For a good separation of the signals from the lower layers ofmulti-layer stacks from the top surface signal in the interferograms, atool z-response function h(z) that is very localized in z and does nothave any appreciable side-lobes is preferred if not required. This iscommon in filter design, (F. J. Harris, On the Use of Windows forHarmonic Analysis with the Discrete Fourier Transform, PROC. IEEE 66,51-83, 1978), where a desired spectral behavior is obtained by designinga filter in the time domain. Here the point of view is reversed and afilter is designed in the frequency domain to obtain the desiredbehavior in the z-domain. A window W(k) is selected as desired toolfunction. Then a calibration measurement is carried out with H(k) asresult. From that the filter function F(k) is determined as:F(k)=W(k)/H(k)  (16)When this filter function is used to modify the light spectrum, theresulting tool function becomes W(k). Examples of advantageous windowfunctions are the Blackman-Harris window or the Dolph-Chebysheff windowmentioned above in the Harris article. For the programmable filters inthe spectral shaping device according to FIGS. 3,4 or 5, the filterfunction F(k) is translated into the necessary transmittance T(x) orreflectance R(x) depending on the chosen optical configuration. Becauseonly the common-path properties of the interferometer are affected, onlythe amplitude of the tool function is affected. Furthermore, the filterfunction is passive, i.e., we can only have F(k)<1. Thus, the support ofthe window function W(k) cannot be larger than the support of theintrinsic tool function H(k). However, it is desirable to minimizemainly the side lobes in the z-domain, not decrease significantly theoverall width of the tool z-response function. Because the presence ofside lobes is largely affected by the smoothness and symmetry of thespectrum, a realizable window function can be employed with significantperformance improvements in the data processing.

Small changes to the tool function H(k) can also be carried out bynumerical filtering. For that purpose, the tool function is determinedaccording to equation (9), and the filter function F(k) is determinedaccording to equation (16). For each camera pixel, the measuredintensity signal is then Fourier-transformed according to equation (7),multiplied with the filter function F(k), and inverselyFourier-transformed to obtain the modified intensity signal i′(z). Thedata processing steps described below then can be carried out with thismodified signal i′(z) as input.

The processing performed in accordance with embodiments and methods ofthe invention may vary from application to application. To betterillustrate the principles of the invention according to these aspects,an exemplary processing scheme for the presently preferred embodimentsand preferred methods as described herein above is shown in FIG. 9. Fora given tool status, the calibration measurement (33) is executed,preferably once, to obtain the tool function H(k) (34). The toolfunction H(k) (34) and the nominal surface model parameters (35) areused to determine the choice of the appropriate weighting windowfunction (36) for the mixed-domain comparison. For each test surfacemeasurement, the intensity profiles at all pixels of interest areacquired (37). At a number of selected camera or detector pixels, aphase reference surface is generated (38). At each pixel of interest,the following process is then executed. From the phase reference surface(38), an initial z position for the modeling determined (39). From theassumed nominal surface model parameters (35), a surface model (40) isgenerated. From the tool function H(k) (34), the surface model (40) andthe initial z position (39), the intensity profile i_(mod)(z) iscalculated (41). The phase of this intensity profile at the z positionis calculated (42), a correction is applied to the initial z position,and the final modeling z position is obtained (39). Then a new intensityprofile i_(mod)(z) (41) is generated from (34), (39), and (40). Fromthis intensity profile i_(mod)(z), the modeled spectrogram or scalogram(43) is obtained. From the measured intensity profile (37) the measuredspectrogram or scalogram (44) can be obtained. Taking the weightingwindow function (36) into account, the measured (44) and modeled (43)spectrograms are compared to find the best match (45). This is done byan iterative non-linear fit, wherein the model parameters (40) arechanged and a new model spectrogram (43) is calculated and compared tothe measurement until a model is found that matches the measurementwell, or a model is found that optimizes the match with the testmeasurement. From the best matching model, the surface parametersconstituting this model, as well as the z position of the top surface(46), are reported as the result.

This processing may be, and preferably is, carried out using aprocessing means such as computer 17, operating under the control ofsoftware in the form of one or more computer programs. In accordancewith still another aspect of the invention, a machine readable medium isprovided that comprises a program, e.g., one or more computer programs,for performing the processing as described herein. The machine readablemedium may be any medium capable of being read by the processing meansaccording to this invention, including but not limited to any of thestorage means or devices described herein above, and/or removable orportable storage devices, such as diskettes, external hard drives, jumpdrives, CDs, DVDs, and the like.

Additional advantages and modifications will readily occur to thoseskilled in the art. Therefore, the invention in its broader aspects isnot limited to the specific details, representative devices and methods,and illustrative examples shown and described. Accordingly, departuresmay be made from such details without departing from the spirit or scopeof the general inventive concept as defined by the appended claims andtheir equivalents.

1. A method for analyzing surface characteristics of a test object usingbroadband light scanning interferometry, the method comprising:acquiring data from a scanning interferometry signal from a surfacelocation of a test object; transforming the acquired data to ann-dimensional function comprising an independent frequency variable andan independent time variable; and comparing the transformedn-dimensional function with a set of models to determine a best matchfrom the comparison.
 2. A method according to claim 1, wherein n equals2.
 3. A method according to claim 1, wherein: the scanninginterferometer comprises an illuminating numerical aperture and animaging numerical aperture; and the acquiring of the data comprisesusing an illuminating numerical aperture that is smaller than theimaging numerical aperture.
 4. (canceled)
 5. A method according to claim1, wherein the transforming of the acquired data to the n-dimensionalfunction comprises creating a spectrogram modified by a. weightingfunction corresponding to desired parameters of interest.
 6. (canceled)7. (canceled)
 8. (canceled)
 9. (canceled)
 10. (canceled)
 11. A methodaccording to claim 1, wherein the method further comprises: dividing thetime variable signal into a plurality of time-differentiated segments;dividing the frequency variable signal into a plurality of timedifferentiated segments corresponding to the plurality of timedifferentiated segments of the time domain signal; and transforming ofthe acquired data into the n-dimensional function comprises using theplurality of time differentiated segments of the frequency variablesignal and the plurality of time differentiated segments of the timevariable signal to construct the n-dimensional function as a function oftime.
 12. A method according to claim 11, wherein the plurality of timedifferentiated segments of the time variable signal overlap one another.13. A method according to claim 11, wherein the plurality of timedifferentiated segments of the frequency variable signal overlap oneanother.
 14. (canceled)
 15. A method according to claim 1 wherein: theacquiring of the data from the scanning interferometry signal comprisesusing light having an original light spectrum; and the method furthercomprises modifying the original light spectrum to facilitate thecomparing of the n-dimensional function with the set of models todetermine the best match from the comparison.
 16. (canceled) 17.(canceled)
 18. (canceled)
 19. A method according to claim 1, furthercomprising using a processor to modify the interferometry signal tofacilitate the comparing of the n-dimensional function with the models.20. A method for analyzing surface characteristics of a test objectusing broadband light scanning interferometry, the method comprising:acquiring data from a scanning interferometry signal from a surfacelocation of a test object; transforming the acquired data to ann-dimensional function having an independent scale variable and anindependent time variable; and comparing the transformed n-dimensionalfunction with a set of stored n-dimensional function models to determinea best match from the comparison.
 21. A method as recited in claim 20,wherein n equals
 2. 22. A method according to claim 20, wherein: thescanning interferometer comprises an illuminating numerical aperture andan imaging numerical aperture; and the acquiring of the data comprisesusing an illuminating numerical aperture that is smaller than theimaging numerical aperture.
 23. A method according to claim 22, whereinthe illuminating numerical aperture is less than about 0.15.
 24. Amethod according to claim 20, wherein the transforming of the acquireddata to the n-dimensional function comprises creating a scalogrammodified by a weighting function corresponding to desired parameters ofinterest.
 25. A method according to claim 20, wherein the transformingof the acquired data to then-dimensional function comprises using atime-scale transform.
 26. A method according to claim 20, wherein thetransforming of the acquired data to then-dimensional function comprisesusing a wavelet transform.
 27. (canceled)
 28. (canceled)
 29. (canceled)30. A method according to claim 20, wherein the method furthercomprises: dividing the time domain signal into a plurality oftime-differentiated segments; dividing the scale variable into aplurality of time differentiated segments corresponding to the pluralityof time differentiated segments of the time variable signal; andtransforming of the acquired data into the n-dimensional functioncomprises using the plurality of time differentiated segments of thescale variable signal and the plurality of time differentiated segmentsof the time variable to construct the n-dimensional function as afunction of time.
 31. A method according to claim 30, wherein theplurality of time differentiated segments of the time variable overlapone another.
 32. A method according to claim 30, wherein the pluralityof time differentiated segments of the scale variable overlap oneanother.
 33. A method according to claim 20, wherein the acquiring ofthe data from the scanning interferometry signal comprises spectralshaping of the interferometry signal.
 34. A method according to claim20, wherein: the acquiring of the data from the scanning interferometrysignal comprising using light having an original light spectrum; and themethod further comprises modifying the original light spectrum tofacilitate the comparing of the n-dimensional function with the set ofstored models.
 35. A method according to claim 34 wherein modifying theoriginal light spectrum comprises using an optical filter comprising aselected transmittance function.
 36. A method according to claim 35wherein the optical filter comprises a programmable filter.
 37. A methodaccording to claim 20, further comprising using a processor to modifythe interferometry signal to facilitate the comparing of then-dimensional function with the models.
 38. A method for analyzing asurface of a test object, the method comprising: using an interferometrysystem comprising a broadband light beam to scan the surface of the testobject and to thereby generate an interferometry signal comprising atime domain signal; applying a time differentiated function to theinterferometry signal to obtain a frequency domain signal; andconstructing a multi-domain function from the frequency domain signaland the time domain signal.
 39. A method according to claim 38, whereinthe multi-domain function comprises a 2-domain function.
 40. (canceled)41. (canceled)
 42. A method according to claim 38, wherein: the using ofthe interferometry system to scan the surface of the test objectcomprises vertically scanning a position on the surface of the testobject and generating the interferometry signal and the time domainsignal at selected unique times during the vertical scanning; andfurther including the dividing of the time domain signal into aplurality of time-differentiated segments.
 43. A method according toclaim 42, wherein the time-differentiated segments of the plurality oftime-differentiated segments are equally spaced from one another intime.
 44. A method according to claim 42, wherein dividing of the timedomain signal into the plurality of time-differentiated segmentscomprises dividing the vertical scan into a corresponding plurality ofvertical scan segments, wherein the vertical scan segments are of equallength.
 45. A method according to claim 38, wherein the method furthercomprises: dividing the time domain signal into a plurality oftime-differentiated segments; dividing the frequency domain signal intoa plurality of time differentiated segments corresponding to theplurality of time differentiated segments of the time domain signal; andwherein the construction of the multi-domain function comprises usingthe plurality of time differentiated segments of the frequency domainsignal and the plurality of time differentiated segments of the timedomain signal to construct the multi-domain function as a function oftime.
 46. A method according to claim 45, wherein the plurality of timedifferentiated segments of the time domain signal overlap one another.47. A method according to claim 46, wherein the plurality of timedifferentiated segments of the frequency domain signal overlap oneanother.
 48. A method according to claim 38, further comprisingcomparing the multi-domain function with a set of models to determine abest match from the comparison.
 49. A method according to claim 38,wherein: the test object comprises m top layers and n bottom layers; theuse of the interferometry system comprises causing the broadband lightbeam to comprise a spectral composition that facilitates obtaining thefrequency domain signal for the m top layers of the test object whiledisfavoring obtaining the frequency domain signal for the n bottomlayers of the test object.
 50. A method according to claim 49, whereinthe use of the interferometry system comprises causing the broadbandlight beam to comprise a. spectral composition such that the frequencydomain signal for the m top layers of the test object is distinguishablefrom the frequency domain signal for the n bottom layers of the testobject.
 51. A method according to claim 49, wherein: the broadband lightbeam comprises amplitude components; and the causing of the broadbandlight beam to comprise a spectral composition that facilitates obtainingthe frequency domain signal for the m top layers of the test objectwhile disfavoring obtaining the frequency domain signal for the n bottomlayers of the test object comprises modifying the spectral compositionbased on the amplitude components.
 52. A method according to claim 49,wherein the use of the interferometry system to cause the broadbandlight beam to comprise a spectral composition that facilitates obtainingthe frequency domain signal for the m top layers of the test objectwhile disfavoring obtaining the frequency domain signal for the n bottomlayers of the test object comprises causing the broadband light sourceto have an original light spectrum that includes the spectralcomposition.
 53. A method according to claim 49, wherein the use of theinterferometry system to cause the broadband light beam to comprise aspectral composition that facilitates obtaining the frequency domainsignal for the m top layers of the test object while disfavoringobtaining. the frequency domain signal for the n bottom layers of thetest object comprises using spectral shaping optics positioned withinthe broadband light beam to spectrally shape the broadband light beam toobtain the spectral composition.
 54. A method according to claim 49,wherein the use of the interferometry system to cause the broadbandlight beam to comprise the spectral composition that facilitatesobtaining the frequency domain signal for the m top layers of the testobject while disfavoring obtaining the frequency domain signal for the nbottom layers of the test object comprises using means positioned withinthe broadband light beam for spectrally shaping the broadband light beamto obtain the spectral components.
 55. A method according to claim 38,wherein: the test object comprises m top layers and n bottom layers; andthe obtaining of the frequency domain signal from the interferometrysignal comprises using a processor to spectrally shape theinterferometry signal to comprise a spectral composition thatfacilitates obtaining the frequency domain signal for the m top layersof the test object while disfavoring obtaining the frequency domainsignal for the n bottom layers of the test object.
 56. A methodaccording to claim 38, wherein: the test object comprises m top layersand n bottom layers; and the obtaining of the frequency domain signalfrom the interferometry signal comprises using a processor to cause theinterferometry signal to comprise a spectral composition such that thefrequency domain signal for the m top layers of the test object isdistinguishable from the frequency domain signal for the n bottom layersof the test object.
 57. (canceled)
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 59. An apparatus foranalyzing surface characteristics of a test object, the apparatuscomprising: a scanning interferometry system that generates a scanninginterferometry signal from a surface location of a test object; and aprocessor that transforms the acquired data to an n-dimensional functionhaving an independent scale variable and an independent time variable,and that compares the transformed n-dimensional function with a set ofstored n-dimensional function models to determine a best match from thecomparison.
 60. (canceled)
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 63. (canceled)64. (canceled)
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 67. (canceled) 68.(canceled)
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 74. An apparatus for processing an interferometry signalfrom an analysis of a test object by a scanning interferometry system,the apparatus comprising a processor that transforms the acquired datato an n-dimensional function having an independent scale variable and anindependent time variable, and that compares the transformedn-dimensional function with a set of stored n-dimensional functionmodels to determine a best match from the comparison.
 75. (canceled) 76.A medium readable by a machine, the medium comprising a program thatcauses the machine to use an interferometry signal comprising a timedomain signal from a broadband interferometry system, whichinterferometry signal is obtained from a test surface, wherein theprogram causes the machine to use the interferometry signal to generatea frequency domain signal, wherein the program causes the machine to usethe time domain signal and the frequency domain signal to construct amulti-domain function that comprises information from the time domainsignal and the frequency domain signal, and wherein the program causesthe machine to use the multi-domain function to obtain informationuseful in characterizing the surface of the test object.
 77. (canceled)78. (canceled)
 79. (canceled)
 80. (canceled)
 81. (canceled) 82.(canceled)
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 86. (canceled)87. (canceled)
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 90. (canceled) 91.(canceled)
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 93. (canceled)
 94. A method for analyzing asurface of a test object, the method comprising: using an interferometrysystem comprising a broadband light beam to scan the surface of the testobject and to thereby generate an interferometry signal comprising atime domain signal; applying a time differentiated function to theinterferometry signal to obtain a scale domain interferometry signal;and constructing a multi-domain function from the frequency domainsignal and the scale domain signal.
 95. A method according to claim 94wherein the multi-domain function comprises a 2-domain function.
 96. Amethod according to claim 94 wherein: using of the interferometry systemto scan the surface of the test object comprises vertically scanningposition on the surface of the test object and generating theinterferometry signal and the time domain signal at selected uniquetimes during the vertical scanning; and further including: dividing ofthe time domain signal into a plurality of time-differentiated segments.97. A method according to claim 94 wherein the method further comprises:dividing the time domain signal into a plurality of time-differentiatedsegments; dividing the scale domain signal into a plurality of timedifferentiated segments corresponding to the plurality of timedifferentiated segments of the time domain signal; and wherein theconstruction of the multi-domain function comprises using the pluralityof time differentiated segments of the scale domain signal and theplurality of time differentiated segments of the time domain signal toconstruct the multi-domain function as a function of time.
 98. A methodaccording to claim 94 further comprising comparing the multi-domainfunction with a set of models to determine a best match from thecomparison.
 99. A method according to claim 94 wherein: the test objectcomprises m top layers and n bottom layers; the use of theinterferometry system comprises causing the broadband light beam tocomprise a spectral composition that facilitates obtaining the scaledomain signal for the m top layers of the test object while disfavoringobtaining the scale domain signal for the n bottom layers of the testobject.
 100. A method according to claim 94 wherein: the test objectcomprises m top layers and n bottom layers; and the obtaining of thescale domain signal from the interferometry signal comprises using aprocessor to spectrally shape the interferometry signal to comprise aspectral composition that facilitates obtaining the scale domain signalfor the m top layers of the test object while disfavoring obtaining thescale domain signal for the n bottom layers of the test object.
 101. Amethod according to claim 94 wherein: the test object comprises m toplayers and n bottom layers and the obtaining of the scale domain signalfrom the interferometry signal comprises using a processor to cause theinterferometry signal to comprise a spectral composition such that thescale domain signal for the m top layers of the test object isdistinguishable from the scale domain signal for the n bottom layers ofthe test object.